What is Algebra

Objectives

After completing this unit you should be able to:

Define algebra.

Recognize when an algebraic statement is an algebraic term,
expression, or equation.

What is Algebra?

Algebra
is a branch of mathematics that uses mathematical statements to
describe relationships between things that vary over time. These
variables include things like the relationship between supply
of an object and its price. When we use a mathematical statement
to describe a relationship, we often use letters to represent
the quantity that varies, since it is not a fixed amount. These
letters and symbols are referred to as variables.
(See the Appendix One for
a brief review of constants and variables.)

The mathematical statements that describe relationships are
expressed using algebraic terms, expressions, or equations
(mathematical statements containing letters or symbols to represent
numbers). Before we use algebra to find information about these
kinds of relationships, it is important to first cover some basic
terminology. In this unit we will first define terms, expressions,
and equations. In the remaining units in this book we will
review how to work with algebraic expressions, solve equations,
and how to construct algebraic equations that describe a relationship.
We will also introduce the notation used in algebra as we move
through this unit.

Algebraic Terms

The basic unit of an algebraic expression is a term. In general,
a term is either a number or a
product of a number and one or more variables. Below
is the term 3ax.

The numerical part of the term, or
the number factor of the term, is what we refer to as the numerical
coefficient.This
numerical coefficient will take on the sign of the operation in
front of it. The term above contains a numerical coefficient,
which includes the arithmetic sign, and a variable or variables.
In this case the numerical coefficient is 3 and the variables
in the term are a and x. Terms such as xz
may not appear to have a numerical coefficient, but they do. The
numerical coefficient is 1, which is assumed.

Algebraic Expressions

An expression is a meaningful
collection of numbers, variables, and signs, positive or negative,
of operations that must make mathematical and logical sense.
Expressions:

contain any number of algebraic terms

use signs of operationaddition, subtraction, multiplication,
and division.

do not contain an equality sign (=)

An example of an expression is:

3ax + 11wx2y

In an expression, the signs of operation separate it into terms.
The sign also becomes part of the term that it follows. The expression
above contains two terms, the first term is 3ax and
the second term is +11wx2y. The addition
sign separates the two terms. For example, in the expression given
above the plus sign (+) separates the 3ax from 11wx2y
and is also part of the second term. Terms that do not have a
sign listed in front of them are understood to be positive.

Below are several examples that are not expressions.

x +  y

This statement tells us "x plus multiplied by y".
This does not make mathematical or logical sense. This collection
of symbols is nonsense.

y = 2x  1

This statement is not an expression because expressions are not
allowed to contain the equal sign.

NOTE:
The operation of multiplication can be represented by using a
x, , or by placing items to be multiplied in parentheses,
brackets or braces, or in the case of variables, just written
next to one another. The statements axb,
a  b, (a)( b), and ab
are equivalent. In this booklet we will use the latter three representations.

Algebraic Equations

An equation is a mathematical
statement that two expressions are equal.The following three statements are equations:

4 + 5 = 9

x  35 = 56k2 + 3

x + 3 = 15

The first equation, 4 + 5 = 9, contains only numbers; the other
two, however, also contain variables. All three contain two expressions
separated by an equal sign:

When an equation contains variables you will often have to
solve for one of those variables. Using equations to solve for
a variable will be discussed later in this booklet.